There are 9 groups of 2, or I have removed 9 numbers from a total of 27 giving me out an answer of 18. o(27) = 27 - 9 = 18 o(8) = 8 - 4 = 4 I can note a relationship developing, 27 being 3 , and 9 being 3 . Just as 8 being 2 , 4 is 2 . Now I can test this theory on 4 . I can predict that for 4 , because it equals 64, that the function will equal 4 - 4 o64 = 64 - 16 = 48 This equation is wrong, as the o function of 64 is not 48, from this I can conclude that the theory of the relationship only works on prime factors. With this information in mind, I can predict for 5 , that the answer will be 5 - 5 . So for any number (n) to the power of 3 the answer would be n3 - n2. I can now apply this knowledge to predict an equation. For any prime number, to any power. 2 = 16 o16 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 = 8 16 - 8 = 8 2 - 2 3 = 81 81 - 27 = 3 - 3 In words, the function of any prime number (p) to any power (n) will be: p x p ...read more.

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I will further investigate this by placing them in a table. x 2x ?(2x) 1 2 1 2 4 2 3 8 4 4 16 8 5 32 16 This can be expressed in a formula: ?2x=1/2 (2x) We must test this formula to check it is right. e.g. ?21=1/2(21)

the answer is the same as subtracting 1 from the original phi number. This is what I predict will happen with all other prime numbers. Trial results based on my prediction: (5) = 5-1 (5) = 4 (11) = 11-1 (11)

By looking at my formulas I also discovered that all my formulas need the Phi values of other numbers at some point or another. This means that there is no certain formula to find the Phi value of a number without using the Phi values of other numbers.

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= 8 This is true. ?(40) = 8[?(4)] ?(40) = 8 x 2 ?(40) = 16 This too is true. Thus we come to know that this above formula is true. Next we will see if there is any relationship within the values for prime numbers.